To solve this problem, we need to analyze the ratios and changes in the number of beads step by step.
1. Initial Ratio: The initial ratio of blue beads to red beads is given as 7:13. Assume the initial quantities of blue and red beads are [tex]\(7x\)[/tex] and [tex]\(13x\)[/tex] respectively, where [tex]\(x\)[/tex] is a common multiplier.
2. Final Ratio: After Susan removes an equal number of both blue and red beads, the final ratio becomes 1:3. Assume the final quantities of blue and red beads are [tex]\(y\)[/tex] and [tex]\(3y\)[/tex] respectively, where [tex]\(y\)[/tex] is another common multiplier.
3. Equal Beads Removed: The number of beads removed from both colors is equal.
- The number of blue beads removed: [tex]\(7x - y\)[/tex]
- The number of red beads removed: [tex]\(13x - 3y\)[/tex]
Since the removed amounts are the same, we get the equation:
[tex]\[7x - y = 13x - 3y\][/tex]
4. Solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[7x - y = 13x - 3y\][/tex]
[tex]\[7x - y + 3y = 13x\][/tex]
[tex]\[7x + 2y = 13x\][/tex]
Simplify,
[tex]\[2y = 6x\][/tex]
[tex]\[y = 3x\][/tex]
5. Initial Total Beads: The initial total number of beads is:
[tex]\[7x + 13x = 20x\][/tex]
6. Final Total Beads: Substitute [tex]\(y = 3x\)[/tex] into the final number of beads:
- Final blue beads: [tex]\(y = 3x\)[/tex]
- Final red beads: [tex]\(3y = 3(3x) = 9x\)[/tex]
Thus, the final total number of beads is:
[tex]\[3x + 9x = 12x\][/tex]
7. Fraction of Beads Left: The fraction of beads left is given by the ratio of the final total number of beads to the initial total number of beads:
[tex]\[\text{Fraction left} = \frac{12x}{20x} = \frac{12}{20} = \frac{3}{5}\][/tex]
8. Percentage of Beads Left: Convert the fraction to a percentage:
[tex]\[\text{Percentage left} = \left(\frac{3}{5}\right) \times 100 = 60\%\][/tex]
Therefore, 60% of the beads were left in the box.
